One of the main ingredients in Apéry's proof of the irrationality of $\zeta(3)$ is the existence of the fast-converging series:
$$ {\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{{\binom {2k}{k}}k^{3}}}\end{aligned}}}. $$
Despite numerous attempts, no similar expressions were found for other values of the Riemann $\zeta$-function at positive odd integers.
For Catalan's constant, however, we do have such an expression, namely:
$${\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}$$
Why is this not sufficient for applying an Apéry-like method for proving its irrationality?
Maybe the central binomial series I've discovered in arXiv paper https://arxiv.org/abs/1207.3139, namely $$G = -\,\frac{1}{2} \, \sum_{n=1}^\infty{(-1)^n \, \frac{(3n-1)\,8^n}{(2n+1)^3 \, \binom{2n}{n}^3}}$$, can be useful for you. There in that paper I've shown that the convergence rate of my series is better than that you mentioned above! Regards, Fabio